In [1]}, small modifications of the TFD state were studied, introducing a perturbation by acting on one side with an operator \(W(t) = e^{-iHt_w} \mathcal {O} e^{iHt_w}\) , where \(\mathcal {O}\)  is some local operator. This “timefold” operator corresponds to evolving the TFD state at \(t=0\)  back into the past for a time \(t_w\) , inserting \(\mathcal {O}\)  and then evolving forward in time by \(t_w\)  back to \(t=0\)  to produce a modified state. In the limit of large \(t_w\) , the dual of this construction is a geometry with a lightlike shock propagating from the boundary at early times into the black hole. The back-reaction of this shock lengthens the wormhole. This investigation was extended to consider multiple shocks in [2]}.
